You have found the following ages (in years) of all 5 porcupines at your local zoo: $ 4,\enspace 2,\enspace 12,\enspace 3,\enspace 15$ What is the average age of the porcupines at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Answer: Because we have data for all 5 porcupines at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{4 + 2 + 12 + 3 + 15}{{5}} = {7.2\text{ years old}} $ Find the squared deviations from the mean for each porcupine. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $4$ years $-3.2$ years $10.24$ years $^2$ $2$ years $-5.2$ years $27.04$ years $^2$ $12$ years $4.8$ years $23.04$ years $^2$ $3$ years $-4.2$ years $17.64$ years $^2$ $15$ years $7.8$ years $60.84$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{10.24} + {27.04} + {23.04} + {17.64} + {60.84}} {{5}} $ $ {\sigma^2} = \dfrac{{138.8}}{{5}} = {27.76\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{27.76\text{ years}^2}} = {5.3\text{ years}} $ The average porcupine at the zoo is 7.2 years old. There is a standard deviation of 5.3 years.